FBX Export: What does Forward / Up assignment do?
Each software handle axis names/orientation differently. That doesn't change anything while your working into a soft, but can be problematic when you export models and animations.

Z Up / Y Up

In Blender, the vertical axis is called Z (so X and Y define the floor plane). In many other software like Unity, the vertical axis is called Y (so X and Z define the floor plane).

Y Forward / -Y Forward

Once you know which axis means vertical, you need to determine how the two other axis are oriented. 



In Blender if you turn the view so the Z axis points up, and the X Axis points right, then the Y axis points backward ; in other terms, the -Y axis points forward.

Sometime this kind of orientation is called "left-handed coordinates" :


Of course it opposes to right-handed coordinates where the "Y-finger" would point forward.

Blender / Unity

Blender is Z Up / -Y Forward

Unity is Y Up / - Z Forward

Other cases

It's quite conventional that the vertical axis points up (not down), and the X axis points right (not left). However, that's not universal, so the fbx exporter allow every valid Up/Forward combination. (an invalid combination would be something paradoxal like : Z Up, Z Forward).